# Parameter uncertainty in curve fitting

My real problem has a much more complexity and a different function than following. However, for the sake of simplicity assume I have a data that can be described as a one dimensional Gaussian function:

$$y = Ae\left(\frac{-(x-\mu)^2}{\sigma}\right)$$

The $$x$$ variable which is the input to this Gaussian function is not an independent variable, rather $$x$$ values could be time for example. The output of the function is only the intensity of a signal.

After I fit my Gaussian function to my data I will have three fitted parameters $$A$$, $$\mu$$, and $$\sigma$$.

If the problem was regression, that is having an independent variable $$x$$ and the dependent variable $$y$$, then I would use the covariance matrix.

However, in this case of curve fitting, $$x$$ is not an independent variable.

How can I estimate the uncertainty in the model fits?

Some references to learn about uncertainty in curve fitting is also appreciated.

• Could you clarify the definition of independent variable you're using here and indicate which aspect of that definition your x-variable doesn't satisfy? It may be important that we're all on the same page with this. Commented May 5 at 23:03